The Conjecture of Dixmier

In Mathematics there are two old open problems: the Jacobian
Conjecture (open since 1938) for the polynomial algebras
in n variables and the
Conjecture of Dixmier (open since 1968) for the algebras A(n) of
polynomial differential operators, the so-called Weyl algebras,
that claims that
the Weyl algebras behave like the finite fields. More precisely,
every algebra endomorphism of the Weyl algebra is an
automorphism. In 1982, Bass, Connell and Wright proved that the
Conjecture of Dixmier implies the Jacobian Conjecture. In
2005-07, Tsuchimoto, Belov-Kanel and Kontsevich proved that these
two conjectures are equivalent. The Weyl algebra A(n) is a
subalgebra of the algebra I(n) of polynomial integro-differential
operators. At the end of 2010, I proved that an an analogue of the
Conjecture of Dixmier holds for the algebra I(1) (V. Bavula, ``An
analogue of the Conjecture of Dixmier is true for the algebra of
polynomial integro-differential operators,'' Arxiv:math.RA:
1011.3009), and conjectured that the same result is true for all
algebras I(n). The aim of this project is to prove this conjecture
and as a result to have a progress on the Conjecture of Dixmier.
Another goal of the project is to find the K-groups for the
algebras I(n) and to answer the question of whether or not the
Bott periodicity holds. The most interesting (and difficult) is
the case of the K(1)-groups for the algebras I(n) since it leads
to finding explicit generators for the automorphism groups of the
algebras I(n). The groups of automorphisms of the algebras I(n)
are infinite dimensional algebraic groups. Little is known about
their structure in general. In the polynomial case there are
several papers by Shafarevich (1966, 1981) and more recently by
Kambayashi (1996, 2003, 2004). We are going to obtain
generalizations of these results for the Weyl algebras A(n) and
I(n).

Who will benefit from the research?

Beneficiaries of results of the project are Pure and Applied
Mathematicians, Physicists and Computer Scientists, or more
generally all who deal with differential and integro-differential
operators.

How they will benefit from the research?

By applying the results and techniques that will be developed in
the project.

What will be done that they benefit from this research?

Talks will be given by the PI and the two postgraduate students at
the conferences MSRI (Berkley) Programs on ``Commutative
Algebra'' (2012) and ``Noncommutative Algebraic Geometry and
Representation Theory'' (2013), by the PI at the University of
Cornell, the Rutgers, the Wayne State University, the University
of Moscow and others, at the School of Mathematics Colloquium and
at the Physics Seminar at the University of Sheffield.